Mesh disappears when removing parent armature relation
Did you notice the mesh dimensions in the 3D view sidebar? It goes from meters and centimeters to centimeters and millimeters. It's not disappearing, but shrinking to such tiny dimensions you can't see it with the armature visible.

Instead of removing the parent at the Object panel, try the following: With the mesh selected and cursor on the 3D view, press Space, then Clear Parent > Clear and Keep Transformation.

That should preserve the transformations affecting the mesh while it was parented to the armature, what won't happen if you simply remove the parent from the panel.